Theorem ralrab2 2757

Description: Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypothesis

Ref	Expression
ralab2.1	|- ( x = y -> ( ps <-> ch ) )

Assertion

Ref	Expression
ralrab2	|- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) )

Distinct variable groups:   x, y   x, A   ch, x   ph, x   ps, y

Allowed substitution hints:   ph( y)   ps( x)   ch( y)   A( y)

Proof of Theorem ralrab2

Step	Hyp	Ref	Expression
1		df-rab 2357	. . 3 |- { y e. A | ph } = { y | ( y e. A /\ ph ) }
2	1	raleqi 2553	. 2 |- ( A. x e. { y e. A | ph } ps <-> A. x e. { y | ( y e. A /\ ph ) } ps )
3		ralab2.1	. . 3 |- ( x = y -> ( ps <-> ch ) )
4	3	ralab2 2756	. 2 |- ( A. x e. { y | ( y e. A /\ ph ) } ps <-> A. y ( ( y e. A /\ ph ) -> ch ) )
5		impexp 259	. . . 4 |- ( ( ( y e. A /\ ph ) -> ch ) <-> ( y e. A -> ( ph -> ch ) ) )
6	5	albii 1399	. . 3 |- ( A. y ( ( y e. A /\ ph ) -> ch ) <-> A. y ( y e. A -> ( ph -> ch ) ) )
7		df-ral 2353	. . 3 |- ( A. y e. A ( ph -> ch ) <-> A. y ( y e. A -> ( ph -> ch ) ) )
8	6, 7	bitr4i 185	. 2 |- ( A. y ( ( y e. A /\ ph ) -> ch ) <-> A. y e. A ( ph -> ch ) )
9	2, 4, 8	3bitri 204	1 |- ( A. x e. { y e. A | ph } ps <-> A. y e. A ( ph -> ch ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   e. wcel 1433   {cab 2067   A.wral 2348   {crab 2352

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rab 2357

This theorem is referenced by: (None)